MATH 231 Lecture 19: Power Series as Functions
8 Apr 2015
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Math 231- lecture 19- power series as functions. We can represent functions as the sum of power series by manipulating geometric series or by differentiating/integrating the series. We can observe that the function equals the sum of a geometric series where a = 1 and r = x x 2 1. We can now view this geometric series as a power series where a =0 and cn=1. We can view many functions in this way if we manipulate them to resemble the general form of. The following theorem allows us to make conclusions about the function based on the radius of convergence of its corresponding power series. C n and have its radius of convergence be greater than zero- n=0. Then f(x) is continuous, differentiable, and integratable for all values of x where. The term by term integration of the function can be calculated using the following sum: x a n+1 f (x)dx=c+ .
